Think of it this way: Anyone can invent a new song, and no matter how derivative or lacking in artistic merit, they can still claim to have composed a new piece of music.(La-la-la-laaa. There, I just did it! Short but sweet.)
But you cannot just "invent" a new proof to the Pythagorean theorem. Any proof that a^2 + b^2 = c^2 will have to clear the hurdle of being demonstrably "true."
In my two semesters of college math, I've gathered that the faculty has something of a phobia to, if not wolfram in particular, students' access to help outside of the department.
Homework problems were oftentimes deliberately difficult, and attending tutoring/office hours was almost certainly necessary for most students to master the material.
I got my hands on an instructor's manual of the textbook, and it was a tremendous boon for my mastery of the topics. By having immediate access to the solutions of difficult problems, I was able to comprehend how to approach problems of that type, and therefore could solve more difficult but similar examples in the future. The cycle of attempt/fail/check-solution/repeat was really effective. Waiting for the instructor's office hours or the availability of tutors would have made this process, if not impossible, incredibly inefficient.
Do any math educators have any insight to this? Is this math department clinging to an antiquated curriculum in which faculty is something of a gate-keeper to knowledge? Is there a good reason for their distaste for 'going around' them?
> the faculty has something of a phobia to ... students' access to help outside of the department.
Math professor here. I am most certainly happy if my students get help outside the department, and I think my attitude is quite typical.
We can be a little bit wary of some kinds of help. Too much math teaching consists of "If you see a problem that looks exactly like X, here are the steps you should memorize to solve it."
But we don't care per se if you can solve problems of the shape X, Y, or Z. We want you to develop your skills to the point that all of these lie naturally within your skill set, that you could do them even if you've never seen one exactly like that before. As such, some kinds of tutoring can be counterproductive.
But most aren't. In my opinion your professors' attitude was quite foolish. Kudos to you for seizing the initiative and figuring out for yourself how to best learn the material.
There are always going to be some students who want to learn the minimum possible to pass the exam, and who will never work with the material again. Although I do my best to be respectful of such students (indeed, in some circumstances this can be a perfectly rational point of view), my pedagogy is aimed at the student who sees my class as something more than a meaningless hoop.
My way of saying it is that it's great if you get help from any source you can, but it's way too easy to get something that seems helpful (because it makes short-term goals easier to achieve) while being damaging in the long run. It's fantastic if you had the personal discipline to use a solution manual to deepen your understanding, but there are lots of students who will use the solution manual as a copybook—the material in it going, as the saying goes, from page to pen without passing through brain on the way. Since I, as a teacher, don't have a ready way at the beginning of the semester to distinguish the students with your discipline from those without, I'm just going to discourage everyone from using solution manuals—but, as long as your homework solutions aren't copied from it, I don't care much if you go against that advice.
Dear god, the timed drills. I distinctly recall the tremendous anxiety evoked by the ultra-competitive nature of those drills in elementary school, and especially the subsequent feelings of failure when comparing my performance to that of my peers.
It must have been that bitter flavor of failure that first inured me against mathematics...it didn't help that the rest of my K-12 education never even attempted to demonstrate the enormous beauty of maths. In fact, until I discovered calculus on my own terms between high school and college, I understood mathematics to be nothing more than the practice of applying rote formulas to arbitrary equations. There was no rhyme or reason to the quadratic equation...it was just one of many "rules" pulled from the mathematical "rulebook," and math was simply the practice of recognizing when this arbitrary rule applied, and then applying it.
Therefore, for most of my life, math was not seen as a creative or exploratory discipline, and in fact the very opposite: One's ability in math was completely contingent on their memorization of rules and simply following them. It was purely robotic, the domain of tightly-wound, uninspired automatons. It was for squares, not free-spirited creative souls like myself!
Though I am disappointed to think of the heights and wonders I could have visited by now, had I been given a proper introduction at a younger age, I am no less excited by the wonders ahead of me, and the many years I have left to explore them. :)
But you cannot just "invent" a new proof to the Pythagorean theorem. Any proof that a^2 + b^2 = c^2 will have to clear the hurdle of being demonstrably "true."